Complex Color Logic

The difficult three or more color puzzles are almost impossible to do without using both the "X"s and "O"s shown in the Extra Tools section. Example 6 showed how to put "O"s on the puzzle to show squares that could only be black. Furthering that thought, you can use the "X"s to show squares that can only be the other color (in this example, red).

Let's look at this line.

We can put "O"s on the first square for the black clue 1. We can also place them on the final 2 squares for the last black clue 2. Can we do anything else on this line?

On the surface it appears we cannot. We don't have quite enough to fill a square from either of the red clues 3. But let's take a closer look.

The black square could be one of three different clues.

Case 1 - The first clue Let's assume the black square is from the black clue 1. Then all the squares preceding it must be white as shown.

Case 2 - The middle clue Now let's assume the black square is from the middle black clue 2. The square that would complete the block of two can go either after the square:

or before the square:

Regardless of where the second black square goes, we know that the three squares before it MUST be either red or white (the background color). Therefore the squares marked in green CANNOT be black.

Case 3 - The last clue The last possibility is that the black square is from the last black clue 2. If this is true, then the row would be colored as such:

As you can see no matter what clue this black square comes from, the seventh and eighth squares cannot be black and therefore can be filled in with an "X" to signify it can only be red (or background).

Similar logic can be applied to the other side of the black square to yield this final result.

If you want some practice on this, an excellent puzzle that uses this type of logic quite frequently is the puzzle Corn.